An Adjusted Gray Map Technique for Constructing Large Four-Level Uniform Designs

doi:10.1007/s11424-023-1144-x

Journal of Systems Science and Complexity ›› 2023, Vol. 36 ›› Issue (1): 433-456.DOI: 10.1007/s11424-023-1144-x

An Adjusted Gray Map Technique for Constructing Large Four-Level Uniform Designs

ELSAWAH A M^1,2,3, VISHWAKARMA G K⁴, MOHAMED H S⁵, FANG Kai-Tai^1,2,6

1. Department of Statistics and Data Science, Faculty of Science and Technology, Beijing Normal UniversityHong Kong Baptist University United International College, Zhuhai 519087, China;
2. Guangdong Provincial Key Laboratory of Interdisciplinary Research and Application for Data Science, BNU-HKBU United International College, Zhuhai 519087, China;
3. Department of Mathematics, Faculty of Science, Zagazig University, Zagazig 44519, Egypt;
4. Department of Mathematics & Computing, Indian Institute of Technology Dhanbad, Dhanbad 826004, India;
5. College of Transportation and Civil Engineering, Fujian Agriculture and Forestry University, Fuzhou 350002, China;
6. The Key Lab of Random Complex Structures and Data Analysis, The Chinese Academy of Sciences, Beijing 100190, China

Received:2021-05-07 Revised:2021-08-22 Online:2023-01-25 Published:2023-02-09
Supported by:
Elsawah’s work was supported by the UIC Research Grants with No. of (R201912 and R202010); the Curriculum Development and Teaching Enhancement with No. of (UICR0400046-21CTL); the Guangdong Provincial Key Laboratory of Interdisciplinary Research and Application for Data Science, BNU-HKBU United International College with No. of (2022B1212010006); and Guangdong Higher Education Upgrading Plan (2021-2025) with No. of (UICR0400001-22).

PDF ( 4 ) Cited

ELSAWAH A M, VISHWAKARMA G K, MOHAMED H S, FANG Kai-Tai. An Adjusted Gray Map Technique for Constructing Large Four-Level Uniform Designs[J]. Journal of Systems Science and Complexity, 2023, 36(1): 433-456.

A uniform experimental design (UED) is an extremely used powerful and efficient methodology for designing experiments with high-dimensional inputs, limited resources and unknown underlying models. A UED enjoys the following two significant advantages: (i) It is a robust design, since it does not require to specify a model before experimenters conduct their experiments; and (ii) it provides uniformly scatter design points in the experimental domain, thus it gives a good representation of this domain with fewer experimental trials (runs). Many real-life experiments involve hundreds or thousands of active factors and thus large UEDs are needed. Constructing large UEDs using the existing techniques is an NP-hard problem, an extremely time-consuming heuristic search process and a satisfactory result is not guaranteed. This paper presents a new effective and easy technique, adjusted Gray map technique (AGMT), for constructing (nearly) UEDs with large numbers of four-level factors and runs by converting designs with $s$ two-level factors and $n$ runs to (nearly) UEDs with $2^{t-1}s$ four-level factors and $2^tn$ runs for any $t\geq0$ using two simple transformation functions. Theoretical justifications for the uniformity of the resulting four-level designs are given, which provide some necessary and/or sufficient conditions for obtaining (nearly) uniform four-level designs. The results show that the AGMT is much easier and better than the existing widely used techniques and it can be effectively used to simply generate new recommended large (nearly) UEDs with four-level factors.

Key words: Aberration, adjusted Gray map technique, Gray map technique, Hamming distance, moment aberration, threshold accepting algorithm, uniform design

[1] Fang K T, The uniform design:Application of number-theoretic methods in experimental design, Acta Math. Appl. Sinica, 1980, 3:363-372.
[2] Wang Y and Fang K T, A note on uniform distribution and experimental design, Chin. Sci. Bull., 1981, 26:485-489.
[3] Hickernell F J, A generalized discrepancy and quadrature error bound, Math. Comp., 1998, 67:299-322.
[4] Hickernell F J, Lattice Rules:How Well Do They Measure Up? Random and Quasi-Random Point Sets, Eds. by Hellekalek P and Larcher G, Springer, New York, 1998.
[5] Zhou Y D, Ning J H, and Song X B, Lee discrepancy and its applications in experimental designs, Statist. Probab. Lett., 2008, 78:1933-1942.
[6] Elsawah A M, Designing uniform computer sequential experiments with mixture levels using Lee discrepancy, Journal of Systems Science and Complexity, 2019, 32(2):681-708.
[7] Elsawah A M, An appealing technique for designing optimal large experiments with three-level factors, J. Computational and Applied Mathematics, 2021, 384:113164.
[8] Lan W G, Wong M K, Chen N, et al., Four-level orthogonal array design as a chemometric approach to the optimization of polarographic reaction system for phosphorus determination, Talanta, 1994, 41(11):1917-1927.
[9] Edmondson R N, Agricultural response surface experiments in view of four-level factorial designs, Biometrics, 1991, 47(4):1435-1448.
[10] Ankenman B E, Design of experiments with two-and four-level factors, J. Quality Technol., 1999, 31(4):363-375.
[11] Phadke M S, Design optimization case studies, AT & T Techn. J., 1986, 65:51-68.
[12] Elsawah A M, Constructing optimal router bit life sequential experimental designs:New results with a case study, Commun. Statist. Simul. Comput., 2019, 48(3):723-752.
[13] Elsawah A M, Designing optimal large four-level experiments:A new technique without recourse to optimization softwares, Communications in Mathematics and Statistics, 2022, 10:623-652.
[14] Bettonvil B and Kleijnen J P C, Searching for important factors in simulation models with many factors:Sequential bifurcation, European J. Oper. Res., 1996, 96:180-194.
[15] Kleijnen J P C, Ham G V, and Rotmans J, Techniques for sensitivity analysis of simulation models:A case study of the CO₂ greenhouse effect, Simulation, 1992, 58(6):410-417.
[16] Kleijnen J P C, Bettonvil B, and Persson F, Screening for the important factors in large discreteevent simulation:Sequential bifurcation and its applications, Screening Methods for Experimentation in Industry, Drug Discovery, and Genetics, Eds. by Dean A and Lewis S, Springer, New York, 2006.
[17] Morris M D, Factorial sampling plans for preliminary computational experiments, Technometrics, 1991, 33:161-174.
[18] Phoa F K H and Xu H, Quarter-fraction factorial designs constructed via quaternary codes, Ann. Statist., 2009, 37:2561-2581.
[19] Phoa F K H, A code arithmetic approach for quaternary code designs and its application to (1/64)th fraction, Ann. Statist., 2012, 40:3161-3175.
[20] Chatterjee K, Ou Z, Phoa F K H, et al., Uniform four-level designs from two-level designs:A new look, Statist. Sinica, 2017, 27:171-186.
[21] Elsawah A M and Fang K T, New results on quaternary codes and their Gray map images for constructing uniform designs, Metrika, 2018, 81(3):307-336.
[22] Hu L, Ou Z, and Li H, Construction of four-level and mixed-level designs with zero Lee discrepancy, Metrika, 2020, 83:129-139
[23] Winke P and Fang K T, Optimal U-Type Designs. Monte Carlo and Quasi-Monte Carlo Methods, Eds. by Niederreiter H, Hellekalek P, Larcher G, and Zinterhof P, Springer, New York, 1997.
[24] Fang K T, Ke X, and Elsawah A M, Construction of uniform designs via an adjusted threshold accepting algorithm, J. Complexity, 2017, 43:28-37.
[25] Elsawah A M and Qin H, Optimum mechanism for breaking the confounding effects of mixed-level designs, Computational Statistics, 2017, 32(2):781-802.
[26] Yang F, Zhou Y D, and Zhang X R, Augmented uniform designs, J. Statist. Plan. Infer., 2017, 182:61-73.
[27] Elsawah A M, Constructing optimal asymmetric combined designs via Lee discrepancy, Statist. Probab. Lett. 2016, 118:24-31.
[28] Tang Y and Xu H, An effective construction method for multi-level uniform designs, J. Statist. Plan. Infer., 2013, 143:1583-1589.
[29] Elsawah A M, Fang K T, and Ke X, New recommended designs for screening either qualitative or quantitative factors, Statistical Papers, 2021, 62:267-307.
[30] Yang F, Zhou Y D, and Zhang A J, Mixed-level column augmented uniform designs, J. Complexity, 2019, 53:23-39.
[31] Elsawah A M, Fang K T, He P, et al., Optimum addition of information to computer experiments in view of uniformity and orthogonality, Bulletin of the Malaysian Math. Sci. Soc., 2019, 42(2):803-826.
[32] Fang K T and Hickernell F J, The uniform design and its applications, Bull. Inst. Int. Stat., 1995, 1:333-349.
[33] Elsawah A M and Qin H, A new strategy for optimal foldover two-level designs, Statist. Probab. Lett., 2015, 103:116-126.
[34] Elsawah A M, Multiple doubling:A simple effective construction technique for optimal two-level experimental designs, Statistal Papers, 2021, 62(6):2923-2967.
[35] Mukerjee R and Wu C F J, On the existence of saturated and nearly saturated asymmetrical orthogonal arrays, Ann. Statist., 1995, 23(6):2102-2115.
[36] Cheng C S, Deng L Y, and Tang B, Generalized minimum aberration and design efficiency for nonregular fractional factorial designs, Statist. Sinica, 2002, 12:991-1000.
[37] Elsawah A M, Building some bridges among various experimental designs, J. Korean Statist. Soc., 2020, 49:55-81.
[38] Xu H, Minimum moment aberration for nonregular designs and supersaturated designs, Statist Sinica, 2003, 13:691-708.
[39] Elsawah A M, Fang K T, He P, et al., Sharp lower bounds of various uniformity criteria for constructing uniform designs, Statistal Papers, 2021, 62:1461-1482.
[40] Cheng C S, Projection Properties of Factorial Designs for Factor Screening Screening, Eds. by Dean A and Lewis S, Springer, New York, 2006.
[41] Sun F, Wang Y, and Xu H, Uniform projection designs, Ann. Statist., 2019, 47(1):641-661.
[42] Elsawah A M, Tang Y, and Fang K T, Constructing optimal projection designs, Statistics, 2019, 53(6):1357-1385.
[43] Elsawah A M and Qin H, An effective approach for the optimum addition of runs to three-level uniform designs, J. Korean Statist. Soc., 2016, 45(4):610-622.
[44] Weng L C, Elsawah A M, and Fang K T, Cross-entropy loss for recommending efficient fold-over technique, Journal of Systems Science and Complexity, 2021, 34(1):402-439.

[1]	Hong QIN, Na ZOU, Shangli ZHANG. DESIGN EFFICIENCY FOR MINIMUM PROJECTION UNIFORMITY DESIGNS WITH TWO LEVELS [J]. Journal of Systems Science and Complexity, 2011, 24(4): 761-768.
[2]	Shixin ZHU;Xiaoshan KAI. THE HAMMING DISTANCES OF NEGACYCLIC CODES OF LENGTH $2^s$ OVER $GR( {2^a,m}) [J]. Journal of Systems Science and Complexity, 2008, 21(1): 60-066.
[3]	Zhao Zhi ZHANG. FURTHER RESULTS ON THE ASYMPTOTIC BEHAVIOUR OF MINIMUM AVERAGE HAMMING DISTANCE FOR BINARY CODES [J]. Journal of Systems Science and Complexity, 1999, 12(4): 335-337.

Viewed

Full text

Abstract